Amortizing randomness in private multiparty computations

We study the relationship between the number of rounds needed to repeatedly perform a private computation (i.e., where there are many sets of inputs sequentially given to the players on which the players must compute a function privately) and the overall randomness needed for this task. For the xor function we show that, by re-using the samerandom bits, we can significantly speed up the round-complexity of each computation compared to what is achieved by the naive strategy of partitioning therandom bits between the computations. Moreover, we prove that our protocols are optimal in the amount of randomness they require.

[1]  Ran Canetti,et al.  Bounds on tradeoffs between randomness and communication complexity , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[2]  Rafail Ostrovsky,et al.  Reducibility and completeness in multi-party private computations , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[3]  Aravind Srinivasan,et al.  Randomness-optimal unique element isolation, with applications to perfect matching and related problems , 1993, STOC '93.

[4]  Eyal Kushilevitz,et al.  Private Computations over the Integers , 1995, SIAM J. Comput..

[5]  Eli Upfal,et al.  A time-randomness tradeoff for oblivious routing , 1988, STOC '88.

[6]  Moni Naor,et al.  Amortized Communication Complexity , 1995, SIAM J. Comput..

[7]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[8]  Eyal Kushilevitz,et al.  A Zero-One Law for Boolean Privacy , 1991, SIAM J. Discret. Math..

[9]  Alfredo De Santis,et al.  Randomness in Distribution Protocols , 1996, Inf. Comput..

[10]  Eyal Kushilevitz,et al.  A Randomness-Rounds Tradeoff in Private Computation , 1994, SIAM J. Discret. Math..

[11]  David R. Karger,et al.  (De)randomized Construction of Small Sample Spaces in NC , 1997, J. Comput. Syst. Sci..

[12]  Alfredo De Santis,et al.  On the Number of Random Bits in Totally Private Computation , 1995, ICALP.

[13]  Eyal Kushilevitz,et al.  A zero-one law for Boolean privacy , 1989, STOC '89.

[14]  Mihir Bellare,et al.  Randomness in interactive proofs , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[15]  Nimrod Megiddo,et al.  Constructing small sample spaces satisfying given constraints , 1993, SIAM J. Discret. Math..

[16]  Russell Impagliazzo,et al.  How to recycle random bits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[17]  Andrew Chi-Chih Yao,et al.  The complexity of nonuniform random number generation , 1976 .

[18]  Matthew K. Franklin,et al.  Communication complexity of secure computation (extended abstract) , 1992, STOC '92.

[19]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[20]  Giulia Galbiati.  M. J. Fischer: On the Complexity of 2-Output Boolean Networks , 1981, Theor. Comput. Sci..

[21]  Nader H. Bshouty,et al.  On the extended direct sum conjecture , 1989, STOC '89.

[22]  Eyal Kushilevitz,et al.  Randomness in Private Computations , 1997, SIAM J. Discret. Math..

[23]  Judit Bar-Ilan,et al.  Non-cryptographic fault-tolerant computing in constant number of rounds of interaction , 1989, PODC '89.

[24]  Wolfgang J. Paul.  Realizing Boolean Functions on Disjoint sets of Variables , 1976, Theor. Comput. Sci..

[25]  Donald Beaver.  Perfect Privacy For Two-Party Protocols , 1989, Distributed Computing And Cryptography.

[26]  David R. Karger,et al.  (De)randomized construction of small sample spaces in /spl Nscr//spl Cscr/ , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[27]  Eyal Kushilevitz,et al.  Fractional covers and communication complexity , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[28]  Rafail Ostrovsky,et al.  Randomness vs. fault-tolerance , 1997, PODC '97.

[29]  Prabhakar Raghavan,et al.  Memory versus randomization in on-line algorithms , 1994, IBM J. Res. Dev..

[30]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[31]  Leonard J. Schulman.  Sample spaces uniform on neighborhoods , 1992, STOC '92.

[32]  Eyal Kushilevitz,et al.  Privacy and communication complexity , 1989, 30th Annual Symposium on Foundations of Computer Science.

[33]  Andrew Chi-Chih Yao,et al.  Theory and Applications of Trapdoor Functions (Extended Abstract) , 1982, FOCS.

[34]  Prabhakar Raghavan,et al.  Memory Versus Randomization in On-line Algorithms (Extended Abstract) , 1989, ICALP.

[35]  Silvio Micali,et al.  How to play ANY mental game , 1987, STOC.

[36]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[37]  Quentin F. Stout,et al.  Meshes with multiple buses , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[38]  Vijay V. Vazirani,et al.  Random polynomial time is equal to slightly-random polynomial time , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[39]  Manuel Blum,et al.  How to generate cryptographically strong sequences of pseudo random bits , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[40]  Y. Mansour,et al.  On construction of k-wise independent random variables , 1994, STOC '94.

[41]  Douglas R. Stinson,et al.  On the Dealer's Randomness Required in Secret Sharing Schemes , 1994, Des. Codes Cryptogr..

[42]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computations , 1990, STOC '90.

[43]  Nimrod Megiddo,et al.  Constructing Small Sample Spaces Satisfying Given Constraints , 1994, SIAM J. Discret. Math..

[44]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[45]  Rafail Ostrovsky,et al.  Characterizing linear size circuits in terms of privacy , 1996, STOC '96.

[46]  David Chaum,et al.  Multiparty unconditionally secure protocols , 1988, STOC '88.

[47]  Avi Wigderson,et al.  Dispersers, deterministic amplification, and weak random sources , 1989, 30th Annual Symposium on Foundations of Computer Science.

[48]  Eyal Kushilevitz,et al.  A Communication-Privacy Tradeoff for Modular Addition , 1993, Inf. Process. Lett..

[49]  Andrew Chi-Chih Yao,et al.  Protocols for secure computations , 1982, FOCS 1982.

[50]  Avi Wigderson,et al.  Completeness theorems for non-cryptographic fault-tolerant distributed computation , 1988, STOC '88.